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Mirrors > Home > MPE Home > Th. List > chpmatply1 | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.) |
Ref | Expression |
---|---|
chpmatply1.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpmatply1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpmatply1.b | ⊢ 𝐵 = (Base‘𝐴) |
chpmatply1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpmatply1.e | ⊢ 𝐸 = (Base‘𝑃) |
Ref | Expression |
---|---|
chpmatply1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpmatply1.c | . . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
2 | chpmatply1.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | chpmatply1.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | chpmatply1.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2798 | . . 3 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
6 | eqid 2798 | . . 3 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
7 | eqid 2798 | . . 3 ⊢ (-g‘(𝑁 Mat 𝑃)) = (-g‘(𝑁 Mat 𝑃)) | |
8 | eqid 2798 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
9 | eqid 2798 | . . 3 ⊢ ( ·𝑠 ‘(𝑁 Mat 𝑃)) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) | |
10 | eqid 2798 | . . 3 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
11 | eqid 2798 | . . 3 ⊢ (1r‘(𝑁 Mat 𝑃)) = (1r‘(𝑁 Mat 𝑃)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chpmatval 21436 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)))) |
13 | 4 | ply1crng 20827 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
14 | 13 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
15 | crngring 19302 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
16 | eqid 2798 | . . . . 5 ⊢ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) | |
17 | 2, 3, 4, 5, 8, 10, 7, 9, 11, 16 | chmatcl 21433 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
18 | 15, 17 | syl3an2 1161 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
19 | eqid 2798 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃)) | |
20 | chpmatply1.e | . . . 4 ⊢ 𝐸 = (Base‘𝑃) | |
21 | 6, 5, 19, 20 | mdetcl 21201 | . . 3 ⊢ ((𝑃 ∈ CRing ∧ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
22 | 14, 18, 21 | syl2anc 587 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
23 | 12, 22 | eqeltrd 2890 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 ·𝑠 cvsca 16561 -gcsg 18097 1rcur 19244 Ringcrg 19290 CRingccrg 19291 var1cv1 20805 Poly1cpl1 20806 Mat cmat 21012 maDet cmdat 21189 matToPolyMat cmat2pmat 21309 CharPlyMat cchpmat 21431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-xor 1503 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-gim 18391 df-cntz 18439 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-dsmm 20421 df-frlm 20436 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-vr1 20810 df-ply1 20811 df-mamu 20991 df-mat 21013 df-mdet 21190 df-mat2pmat 21312 df-chpmat 21432 |
This theorem is referenced by: chmaidscmat 21453 cpmidgsum 21473 cpmidgsumm2pm 21474 cpmidpmatlem2 21476 cpmidpmatlem3 21477 chcoeffeqlem 21490 cayhamlem3 21492 cayleyhamilton1 21497 |
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